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Domains of commutative C*-subalgebras

Published online by Cambridge University Press:  21 March 2019

Chris Heunen and
Bert Lindenhovius
Chris Heunen
Affiliation:
School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, UK
Bert Lindenhovius*
Affiliation:
Department of Computer Science, Tulane University, 6823 St. Charles Avenue, New Orleans, LA 70118, USA
*
*Corresponding author. Email: alindenh@tulane.edu
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Abstract

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A C*-algebra is determined to a great extent by the partial order of its commutative C*-subalgebras. We study order-theoretic properties of this directed-complete partially ordered (dcpo). Many properties coincide: the dcpo is, equivalently, algebraic, continuous, meet-continuous, atomistic, quasi-algebraic or quasi-continuous, if and only if the C*-algebra is scattered. For C*-algebras with enough projections, these properties are equivalent to finite-dimensionality. Approximately finite-dimensional elements of the dcpo correspond to Boolean subalgebras of the projections of the C*-algebra. Scattered C*-algebras are finitedimensional if and only if their dcpo is Lawson-scattered. General C*-algebras are finite-dimensional if and only if their dcpo is order-scattered.

Keywords

C*-algebracommutative C*-subalgebradomainscattered

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Issue 7 , August 2019, pp. 972 - 1006
Copyright
© Cambridge University Press 2019 

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